I begin this lesson by projecting a two-way table and a multiple choice question (see Launch Prompt). I want to access my students prior knowledge of frequency tables. In this exercise students need to make a judgment with respect to using the horizontal (row) or vertical (column) frequencies. The task also requires students to add two conditional frequencies, in order to determine the relative frequency needed to answer the question successfully.

As my students work on the Launch Prompt I listen in on their conversations. I use this time to gain a sense of student understanding. The correct answer is D, 33.3%, because a total of 150 people in that age group (over 40) were surveyed. Of these, 50 were against the increase. Thus, a third (33.3%) of the responents over 40 were against minimum wage increase. I am interested in the process that my students undertake to arrive at this (or another) answer.

For example, several students may mistakingly choose answer B, 71.4%. This answer is a common choice because there are 50 people over 40 years old. Comparing this answer to the 70 respondents that answered against results in the answer 71.4%. When students reason that the correct answer is B, I like to discuss the question for which B is a valid answer. I will say something like, "If the question had been What percent of the people who oppose increasing the minimun wage are over 40?" then answer B, 71.4%, is correct. Can someone tell me why?" (See my Rewinding when Necessary reflection for more on this point.)

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I pair students up by similar ability levels and hand each pair a copy of the Two Way Table Activity. The activity is language intensive, so I avoid pairing two ELL students together. I will check in with groups to make sure that all of my students are reading and trying to comprehend the information on the handout. I expect most of the work on this task to be reasoning about the data using the relative frequencies.

Each handout contains 4 two-way frequency tables with questions probing students' interpretations of the given information. In order to successfully answer the prompts, students need to determine both the marginal and the relative frequencies. ﻿

I find that questions about the interpretation of data provide a good opportunity to assess students' thinking and to encourage students to engage in mathematical practices. So, as the students work, I walk around looking to question students about their answers (correct or incorrect). For this type of work, the students reasoning can be highly diverse, so I often find that the best questions to ask are the ones that occur to me on the spot, as if we are working together on the data.

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To close the lesson I choose the table from the Two Way Table handout that gave the most students trouble and I lead a class discussion about the table. I like to begin this discussion by sharing some of the incorrect interpretations that are possible (including some that I overheard during the lesson) and discussing the validity of the answers.

Why might someone interpret the data this way?

What assumptions might some one make if they had interpreted the data this way?

What miscalculations are present in this answer?

This approach typically pushes my students to think more deeply about the problems. Although we want to discuss the answer and a correct justification for it, when working with data and statistics it is often important to be able to explain why an interpretation is flawed. This skill is an important statistical literacy, and, practicing it helps students to acquire conceptual understanding and avoid making these mistakes.

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Working with two-way frequency tables challenges my students. I find that the more practice I give them, the more they like it. So, I have a quick homework assignment for them. HOMEWORK Lesson 7 should help students to build confidence in their skill at interpreting tables.